background preloader

Optimization

Facebook Twitter

MinFunc - unconstrained differentiable multivariate optimization in Matlab. Mark Schmidt (2005-2012) minFunc is a Matlab function for unconstrained optimization of differentiable real-valued multivariate functions using line-search methods. It uses an interface very similar to the Matlab Optimization Toolbox function fminunc, and can be called as a replacement for this function. On many problems, minFunc requires fewer function evaluations to converge than fminunc (or minimize.m). Further it can optimize problems with a much larger number of variables (fminunc is restricted to several thousand variables), and uses a line search that is robust to several common function pathologies. The default parameters of minFunc call a quasi-Newton strategy, where limited-memory BFGS updates with Shanno-Phua scaling are used in computing the step direction, and a bracketing line-search for a point satisfying the strong Wolfe conditions is used to compute the step direction.

Features Usage minFunc uses an interface very similar to Matlab's fminunc. Example Download Updates. Rate of convergence. In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations.

Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization . Convergence speed for iterative methods[edit] Basic definition[edit] Kendell A.

Wei Fan - Welcome. Software and Datasets related to my work Random Decision Tree (RDT) What it is Random decision tree algorithm constructs multiple decision trees randomly. When constructing each tree, the algorithm picks a "remaining" feature randomly at each node expansion without any purity function check (such as information gain, gini index, etc.). A tree stops growing any deeper if one of the following conditions is met: A node becomes empty or there are no more examples to split in the current node. Each node of the tree records class distributions. The algorithm does not prune the randomly built decision tree in the conventional sense (such as MDL-based pruning and cost-based pruning, and etc.). Classification is always done at the leaf node level. In some situations, a leaf node could be empty. In order to make a final prediction, a loss function is needed. Theoretical Explanation Random decision tree is an efficient implementation of Bayes Optimal Classifier or (OBC).

Examples target distribution. Unconstrained optimization: L-BFGS and CG - ALGLIB. Unconstrained optimization: L-BFGS and CG About algorithms ALGLIB package contains three algorithms for unconstrained optimization: L-BFGS, CG and Levenberg-Marquardt algorithm. This article considers first two algorithms, which share common traits: they solve general form optimization problem (target function has no special structure) they need function value and its gradient only (Hessian is not needed) both algorithms support numerical differentiation when no analytic gradient is supplied by user L-BFGS algorithm L-BFGS algorithm builds and refines quadratic model of a function being optimized.

Essential feature of the algorithm is positive definiteness of the approximate Hessian. Another essential property is that only last M function/gradient pairs are used, where M is moderate number smaller than problem size N, often as small as 3-10. Nonlinear conjugate gradient method Unlike L-BFGS algorithm, nonlinear CG does not build quadratic model of a function being optimized.

Performance. Singular value decomposition. Visualization of the SVD of a two-dimensional, real shearing matrixM. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of M, which distorts the disk to an ellipse. The SVD decomposes M into three simple transformations: an initial rotationV*, a scaling Σ along the coordinate axes, and a final rotation U.

The lengths σ1 and σ2 of the semi-axes of the ellipse are the singular values of M, namely Σ1,1 and Σ2,2. Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form where U is a m×m real or complex unitary matrix, Σ is an m×n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and V* (the conjugate transpose of V, or simply the transpose of V if V is real) is an n×n real or complex unitary matrix.

The singular value decomposition and the eigendecomposition are closely related. Statement of the theorem[edit] The diagonal entries Intuitive interpretations[edit] and.